Finite Group Databasebeta

← #10694616

Subgroups — #10694616

Order 768.

396 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83343C211yesmaximal, characteristic
2C3×#1531C211yesmaximal, characteristic, complemented
3#71836C211yesmaximal, characteristic, complemented
4#70832C211yesmaximal, characteristic
5#70794C211yesmaximal, characteristic, complemented
6#69806C211yesmaximal, characteristic, complemented
7#69798C211yesmaximal, characteristic
8#9766—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C2×C3×#97C2211yescharacteristic
13#5088C411yescharacteristic, complemented
14#5085C411yescharacteristic, complemented
15#5080C411yescharacteristic, complemented
16#5079C411yescharacteristic, complemented
17#5048—21no—
18#5048—21no—
19C4×#215—21no—
20C4×#215—21no—
21#4932C2211yescharacteristic
22#4926C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24C3×#330—22no—
25#4550—21no—
26#4550—21no—
27#4538—22no—
28#4480—21no—
29#4479—21no—
30#4470—22no—
31#4464—22no—
32#3167—31no—
33#1746—31no—
34#1568—31no—
35#1531D311yescharacteristic, complemented
36#1530—31no—
37#1506—31no—
38#1505—31no—
39C2×#235—21no—
40C22×C3×C8—21no—
41C3×C4⋊Q8C2×C411yescharacteristic
42C3×C4⋊Q8C2×C411yescharacteristic
43C3×#125—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×#117—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#215—21no—
51C2×#215—21no—
52#930—21no—
53#930—21no—
54C2×C4×Dic3—21no—
55C22×C3⋊C8D411yescharacteristic
56C22×C3⋊C8D411yescharacteristic
57#875—21no—
58#874C2×C411yescharacteristic
59C4×D12—21no—
60C4×D12—21no—
61#869C2×C411yescharacteristic
62C3×C4⋊C8—21no—
63C3×C22⋊C8—21no—
64C3×#97—21no—
65C3×#97—21no—
66C3×#97D412yes—
67#835—21no—
68#835—21no—
69C3⋊#95—21no—
70C3⋊#95D412yes—
71#830—21no—
72#809—21no—
73#809—21no—
74#808—21no—
75#808—21no—
76C4⋊(C3⋊C8)—21no—
77#803—21no—
78C4×C3⋊C8C2×C411yescharacteristic
79C4×C3⋊C8C2×C411yescharacteristic
80C4×C3⋊C8D411yescharacteristic
81C4×C3⋊C8D411yescharacteristic
82#533C2×D311yescharacteristic
83#532—31no—
84#498—31no—
85#492—31no—
86#487—31no—
87#486—31no—
88#465—61no—
89#465—61no—
90C4×SD16—61no—
91C4×SD16—61no—
92#432—31no—
93#432C2×D311yescharacteristic
94C2×#97C2×D311yescharacteristic
95C2×C4×C8—31no—
96#376—31no—
97#336—61no—
98#336—61no—
99#335—61no—
100#334—61no—
101#330—22no—
102#329—62no—
103#328—62no—
104#327—62no—
105C2×C3×Q8—21no—
106C2×C3×Q8—21no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C2×#67—41no—
111C22×Dic3—22no—
112#235—82no—
113C2×D12—21no—
114C2×D12—21no—
115C2×D12—21no—
116C2×C4×D3—41no—
117C2×Dic6—21no—
118C2×C3×C8—22no—
119C2×C3×C8—41no—
120C2×C3×C8—41no—
121C2×C3×C8—41no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C22⋊C4—41no—
129C3×C42—41no—
130C3×C42C22×C411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42#4111yescharacteristic
134#215—42no—
135#215—42no—
136(C2×D3)⋊C4—41no—
137(C2×D3)⋊C4—41no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4×Dic3—41no—
141C4×Dic3—41no—
142C2×C3⋊C8—41no—
143C2×C3⋊C8—41no—
144C2×C3⋊C8—41no—
145C2×C3⋊C8C22⋊C411yescharacteristic
146C2×C3⋊C8C22⋊C411yescharacteristic
147C2×C3⋊C8C22⋊C411yescharacteristic
148C2×C3⋊C8C22⋊C411yescharacteristic
149C2×#41—61no—
150C2×SD16—61no—
151C2×SD16—61no—
152C22×C8—21no—
153C22×C8—31no—
154C22×C8—31no—
155C4⋊Q8—31no—
156C4⋊Q8C4×D311yescharacteristic
157C4⋊Q8C4×D311yescharacteristic
158#127—31no—
159#125—21no—
160#124—61no—
161C22⋊Q8—21no—
162C22⋊Q8—21no—
163#121—61no—
164#121—61no—
165C4×Q8—21no—
166C4×Q8—21no—
167C4×D4—61no—
168C4×D4—61no—
169#117—21no—
170C2×C42—61no—
171C2×C42C22×D311yescharacteristic
172#106—61no—
173#106—61no—
174C4⋊C8—21no—
175C4⋊C8—61no—
176#103—61no—
177#103—61no—
178#102—61no—
179#102—61no—
180C22⋊C8—21no—
181C22⋊C8—61no—
182#97—21no—
183#97—21no—
184#97—61no—
185#97#6712yes—
186C4×C8—31no—
187C4×C8—31no—
188C4×C8—31no—
189C4×C8—31no—
190#95—32no—
191#95—61no—
192C23×C3#12111yescharacteristic
193C22×D3—41no—
194C3×Q8—41no—
195C3×Q8—41no—
196C3×Q8—81no—
197C2×C3×C4—22no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4C2×C22⋊C411yescharacteristic
204C2×C3×C4C4×D411yescharacteristic
205C2×C3×C4C4×D411yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12411yescharacteristic
208C2×C3×C4#12711yescharacteristic
209#67—82no—
210C2×Dic3—42no—
211C2×Dic3—42no—
212C2×Dic3—41no—
213C2×Dic3—42no—
214D12—41no—
215D12—41no—
216D12—81no—
217D12—81no—
218C4×D3—82no—
219Dic6—81no—
220C3×C8—42no—
221C3×C8—81no—
222C3⋊C8—21no—
223C3⋊C8—21no—
224C3⋊C8—21no—
225C3⋊C8—21no—
226C3⋊C8—81no—
227#41—242no—
228C2×Q8—21no—
229C2×Q8—21no—
230C2×Q8—61no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—121no—
235C22×C4—62no—
236C22×C4—121no—
237C22×C4#23511yescharacteristic
238C22×C4D3×D411yescharacteristic
239C22×C4D3×D411yescharacteristic
240SD16—122no—
241SD16—122no—
242C2×C8—22no—
243C2×C8—31no—
244C2×C8—31no—
245C2×C8—31no—
246C2×C8—31no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—121no—
251C2×C8—121no—
252C2×C8—121no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C22⋊C4—41no—
262C22⋊C4—121no—
263C22⋊C4—121no—
264C42—41no—
265C42—121no—
266C42—121no—
267C42C2×C4×D311yescharacteristic
268C42D3×D411yescharacteristic
269C42#23711yescharacteristic
270C42C2×#6711yescharacteristic
271C22×C3—21no—
272C22×C3—21no—
273C22×C3#39011yescharacteristic
274C22×C3#47112yes—
275C2×D3—81no—
276C2×D3—81no—
277C2×D3—81no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—81no—
284C3×C4—81no—
285C3×C4#42111yescharacteristic
286C3×C4#42111yescharacteristic
287Dic3—82no—
288Dic3—81no—
289C23—121no—
290C23#88411yescharacteristic
291Q8—41no—
292Q8—41no—
293Q8—81no—
294Q8—241no—
295D4—121no—
296D4—121no—
297D4—241no—
298D4—241no—
299D4—242no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—122no—
307C2×C4—122no—
308C2×C4—121no—
309C2×C4—122no—
310C2×C4—242no—
311C2×C4D3×C22⋊C411yescharacteristic
312C2×C4#67⋊C411yescharacteristic
313C2×C4#88511yescharacteristic
314C2×C4C4×#6711yescharacteristic
315C2×C4#93911yescharacteristic
316C2×C4#94011yescharacteristic
317C8—42no—
318C8—61no—
319C8—61no—
320C8—61no—
321C8—61no—
322C8—81no—
323C8—241no—
324C2×C3—22no—
325C2×C3#196311yescharacteristic
326C2×C3#197611yescharacteristic
327C2×C3#197711yescharacteristic
328D3—161no—
329C22—21no—
330C22—21no—
331C22—241no—
332C22—241no—
333C22—241no—
334C22#475412yes—
335C22#495711yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—81no—
342C4—81no—
343C4—242no—
344C4—241no—
345C4#482111yescharacteristic
346C4#482111yescharacteristic
347C3#976611yescharacteristic, complemented, Sylow
348C2—22no—
349C2—481no—
350C2#7010711yescharacteristic
351C2#7035711yescharacteristic
352C2#7039711yescharacteristic
subgroupG/H
centerC22#4957
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1531C2
socleC22×C3#390
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1531D3
hypocenterC3#9766
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1531

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C3×#1531C2semidirect
3. #71836C2semidirect
5. #70794C2semidirect
6. #69806C2semidirect
13. #5088C4semidirect
14. #5085C4semidirect
15. #5080C4semidirect
16. #5079C4semidirect
35. #1531D3semidirect
347. C3#9766semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#833432
C3×#15312
#718362
#708322
#707942
#698062
#697982
#97663